#508794

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,35)(28,37)(29,36)(30,40)(31,41)(32,39)(33,38)(34,42)(43,46)(44,47)(45,48);(2,5)(6,8)(10,13)(14,15)(19,27)(20,30)(21,32)(23,35)(25,40)(26,39)(28,45)(29,44)(31,34)(33,43)(36,47)(37,48)(38,46)(41,42);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,29)(18,23,24,36)(20,31,33,45)(21,34,32,28)(25,41,38,48)(26,42,39,37)(27,43,44,30)(35,46,47,40);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(18,24)(23,36)(25,38)(26,39)(35,47)(37,42)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415990);